On orbital partitions and exceptionality of primitive permutation groups

Let $G$ and $X$ be transitive permutation groups on a set $\Omega$ such that $G$ is a normal subgroup of $X$. The overgroup $X$ induces a natural action on the set $\operatorname{Orbl}(G,\Omega)$ of non-trivial orbitals of $G$ on $\Omega$. In the study of Galois groups of exceptional covers of curves, one is led to characterizing the triples $(G,X,\Omega)$ where $X$fixes no elements of $\operatorname{Orbl}(G,\Omega)$; such triples are called exceptional. In the study of homogeneous factorizations of complete graphs, one is led to characterizing quadruples $(G,X,\Omega,\mathcal{P})$ where $\mathcal{P}$ is a partition of $\operatorname{Orbl}(G,\Omega)$ such that $X$ is transitive on $\mathcal{P}$; such a quadruple is called a TOD (transitive orbital decomposition). It follows easily that the triple $(G,X,\Omega)$ in a TOD $(G,X,\Omega,\mathcal{P})$is exceptional; conversely if an exceptional triple $(G,X,\Omega)$ is such that $X/G$ is cyclic of prime-power order, then there exists a partition $\mathcal{P}$ of $\operatorname{Orbl}(G,\Omega)$ such that $(G,X,\Omega,\mathcal{P})$ is a TOD. This paper characterizes TODs $(G,X,\Omega,\mathcal{P})$ such that $X^\Omega$ is primitive and $X/G$ is cyclic of prime-power order. An application is given to the classification of self-complementary vertex-transitive graphs.